Essential Concepts

• The cross product [latex]{\bf{u}}\times{\bf{v}}[/latex] of two vectors [latex]{\bf{u}}=\langle{u_1,u_2,u_3}\rangle[/latex] and [latex]{\bf{v}}=\langle{v_1,v_2,v_3}\rangle[/latex] is a vector orthogonal to both [latex]{\bf{u}}[/latex] and [latex]{\bf{v}}[/latex]. Its length is given by [latex]\parallel{\bf{u}}\times{\bf{v}}\parallel=\parallel{\bf{u}}\parallel\cdot\parallel{\bf{v}}\parallel\cdot\sin\theta[/latex], where [latex]\theta[/latex] is the angle between [latex]{\bf{u}}[/latex] and [latex]{\bf{v}}[/latex]. Its direction is given by the right-hand rule.
• The algebraic formula for calculating the cross product of two vectors, [latex]{\bf{u}}=\langle{u_1,u_2,u_3}\rangle[/latex] and [latex]{\bf{v}}=\langle{v_1,v_2,v_3}\rangle[/latex], is [latex]{\bf{u}}\times{\bf{v}}=(u_{2}v_{3}-u_{3}v_{2}){\bf{i}}-(u_{1}v_{3}-u_{3}v_{1}){\bf{j}}+(u_{1}v_{2}-u_{2}v_{1}){\bf{k}}[/latex].
• The cross product satisfies the following properties for vectors [latex]{\bf{u}}[/latex], [latex]{\bf{v}}[/latex], and [latex]{\bf{w}}[/latex], and scalar [latex]c[/latex]:
  •  [latex]{\bf{u}}\times{\bf{v}}=-({\bf{v}}\times{\bf{u}})[/latex]
  • [latex]{\bf{u}}\times\left({\bf{v}}+{\bf{w}}\right)={\bf{u}}\times{\bf{v}}+{\bf{u}}\times{\bf{w}}[/latex]
  • [latex]c({\bf{u}}\times{\bf{v}})=(c{\bf{u}})\times{\bf{v}}={\bf{u}}\times(c{\bf{v}})[/latex]
  • [latex]{\bf{u}}\times{0}={0}\times{\bf{u}}=0[/latex]
  • [latex]{\bf{v}}\times{\bf{v}}=0[/latex]
  • [latex]{\bf{u}}\cdot({\bf{v}}\times{\bf{w}})=({\bf{u}}\times{\bf{v}})\cdot{\bf{w}}[/latex]
• The cross product of vectors  [latex]{\bf{u}}=\langle{u_1,u_2,u_3}\rangle[/latex] and [latex]{\bf{v}}=\langle{v_1,v_2,v_3}\rangle[/latex] is the determinant [latex]\left|\begin{array}{ccc} {\bf{i}} & {\bf{j}} & {\bf{k}} \\ u_{1} & u_{2} & u_{3} \\ v_{1} & v_{2} & v_{3}\end{array}\right|[/latex]
• If vectors [latex]{\bf{u}}[/latex] and [latex]{\bf{v}}[/latex] form adjacent sides of a parallelogram, then the area of the parallelogram is given by [latex]\parallel{\bf{u}}\times{\bf{v}}\parallel[/latex].
• The triple scalar product of vectors [latex]{\bf{u}}[/latex], [latex]{\bf{v}}[/latex], and [latex]{\bf{w}}[/latex] is [latex]{\bf{u}}\cdot({\bf{v}}\times{\bf{w}})[/latex].
• The volume of a parallelepiped with adjacent edges given by vectors [latex]{\bf{u}}[/latex], [latex]{\bf{v}}[/latex], and [latex]{\bf{w}}[/latex] is [latex]V=|{\bf{u}}\cdot({\bf{v}}\times{\bf{w}})|[/latex]
• If the triple scalar product of vectors [latex]{\bf{u}}[/latex], [latex]{\bf{v}}[/latex], and [latex]{\bf{w}}[/latex] is zero, then the vectors are coplanar. The converse is also true: If the vectors are coplanar, then their triple scalar product is zero.
• The cross product can be used to identify a vector orthogonal to two given vectors or to a plane.
• Torque [latex]\tau[/latex] measures the tendency of a force to produce rotation about an axis of rotation. If force [latex]{\bf{F}}[/latex] is acting at a distance [latex]{\bf{r}}[/latex] from the axis, then torque is equal to the cross product of [latex]{\bf{r}}[/latex] and [latex]{\bf{F}}[/latex]: [latex]\tau={\bf{r}}\times{\bf{F}}[/latex].